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Math Help - Bounded linear operator

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    Bounded linear operator

    Let X and Y be normed spaces and T_n : X--->Y (n=1,2.....) bounded linear operators. Show that converges T_n---->T implies that for every \epsilon > 0 there is an N such that for all n>N and all x in any given closed ball we have ||T_nx _ Tx||<\epsilon
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kinkong View Post
    Let X and Y be normed spaces and T_n : X--->Y (n=1,2.....) bounded linear operators. Show that converges T_n---->T implies that for every \epsilon > 0 there is an N such that for all n>N and all x in any given closed ball we have ||T_nx _ Tx||<\epsilon
    What have you tried, man? So by definition what does it mean that T_n\to T in the operator norm?
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    yes i tried....but i couldnt proceed
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kinkong View Post
    yes i tried....but i couldnt proceed
    So, you're given some closed ball B_\delta(x) and what you want to prove is that \left\|T_ny-Ty\right\|\to 0 for every [tex]y\in B_\delta(x)[tex], right? Well, you know that \|y\|\leqslant \|x\|+\delta for every y\in B_\delta(x) and so consequently \left\|T_ny-T_y\right\|\leqslant \|T_n-T\|_\text{op}\|y\|\leqslant \|T_n-T\|(\|x\|+\delta)\to0.
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